For \(i \in \mathbb {N}_{n_x}\) , consider any arbitrary \(z^1\)   from the interior of \(\mathbb {I}\mathcal {Z}\)  and construct \(z^2=z^1+\lambda e_j \in \mathbb {I}\mathcal {Z}\) , for some small enough \(\lambda >0\) . Then, by  an identifical proof to [1]}, there exists a \( z_{\theta } \in \mathbb {I}\mathcal {Z}\)  on the connecting line between \(z^1\)  and \(z^2\) , such that \(\lambda f^\downarrow _{i,C}(z_{\theta },e_j)=f^\downarrow _{i,C}(z_{\theta },\lambda e_j) \le f_i(z^2)-f_i(z^1) \le f^\uparrow _{i,C}(z_{\theta },\lambda e_j) =\lambda f^\uparrow _{i,C}(z_{\theta },e_j)\) , where the equalities hold by [1]}. Using this and the CJSS assumption (cf. (REF )), \(f_i(z^2) \le f_i(z_2)\)  if \(f^\uparrow _{i,C}(z,\lambda e_j)=(\overline{J}^f_C(z))_{ij}\le 0\)  holds for all \(z \in \mathbb {I}\mathcal {Z}\)   (including \(z=z_{\theta }\) ), and  similarly, \(f_i(z^2) \ge f_i(z_2)\)  if \(f^\downarrow _{i,C}(z,\lambda e_j)=(\underline{J}^f_C(z))_{ij}\ge 0\)  holds for all \(z \in \mathbb {I}\mathcal {Z}\)   (including \(z=z_{\theta }\) ). Hence, by moving  along the coordinate directions, one can always increase or decrease each of the \(f_i\) 's.  Using a similar argument, this result also holds for \(\lambda <0\) . So, the optimum for each \(f_i\)  is  attained at some vertices of the interval
\(\mathbb {I}\mathcal {Z}\) .
